Search: id:A125294
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%I A125294
%S A125294 1,5,7,5,11,91,1,17,19,11,23,13,1,29,31,17,1,703,1,41,43,23,47,1,1,53,
               1,
%T A125294 29,59,1891,1,1,67,1,71,2701,1,1,79,41,83,43,1,89,1,47,1,97,1,101,103,
%U A125294 53,107,109,1,113,1,59,1,61,1,1,127,1,131,67,1,137,139,71,1,73,1,149
%N A125294 Numerator of Sum[ k^2, {k,1,n} ] / Product[ k^2, {k,1,n} ].
%C A125294 All a(n) are either 1, semiprime or prime. a(n) = 1 for n = 1 and n = 
               {7,13,17,19,24,25,27,31,32,34,37,38,43,45,47,49,...} = A067656 Numbers 
               n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli 
               numbers. p divides a(p-1) for prime p>3. p divides a((p-1)/2) for 
               prime p>3.
%C A125294 a(p-1) = p*(2p-1) is a semiprime hexagonal number for prime p = {7,19,
               31,37,79,97,139,157,199,211,229,271,307,331,337,367,379,439,499,...} 
               = A005382(n) for n>2, where A005382(n) are the numbers n such that 
               n and 2n-1 are primes.
%C A125294 a(p-1) = p for prime p = {5, 11, 13, 17, 23, 29, 41, 43, 47, 53, 59, 
               61, 67, 71, 73, 83, 89, ...} = Primes that do not belong to A005382(n). 
               a((p-1)/2) = p for prime p = {5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 
               47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 
               131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 
               211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 259, 271, 281, 
               283, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 
               379, 383, 389, 401, ...} that is apparently a union of {5} and A034849(n).
%F A125294 a(n) = Numerator[ Sum[ k^2, {k,1,n} ] / Product[ k^2, {k,1,n} ] ]. a(n) 
               = Numerator[ n(n+1)(2n+1)/6/(n!)^2 ].
%t A125294 Table[Numerator[n(n+1)(2n+1)/6/(n!)^2],{n,1,500}]
%Y A125294 Cf. A067656, A005382, A034849.
%Y A125294 Sequence in context: A065478 A109353 A121595 this_sequence A139428 A063005 
               A145577
%Y A125294 Adjacent sequences: A125291 A125292 A125293 this_sequence A125295 A125296 
               A125297
%K A125294 nonn,frac
%O A125294 1,2
%A A125294 Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 17 2007

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